Warning!
The Flying Weapon
When you inflict it about 10000 hp of damage, it will run away. Follow it (if you survived, that is) aboard the Highwind and ram it again. There will be another fight. When it has 10000 (or so) more damage, it will run away again.
You will have to repeat this a few times, like 5 or 6. The last time, the weapon will flee to a spot near Cosmo Canyon (Valley of the Fallen Star, in the Japanese version). When you finally kill it, it will give you the Ultima Weapon for Cloud, then will fall to the ground and create (yet another) crater. If you don't have a Gold Chocobo by then, use it to access the Frog Forest. The Frog Forest is a bonus/candy dungeon, where you will have to solve a couple of puzzles. You will get some materia (Slash-all command materia, the Typoon (was Chupon in FF III and in the Japanese version) summon materia plus a couple of other goodies).
The Flying Weapon must have about 120,000 to 150,000 hp, but it's not an easy kill. When I killed it, it casted Shadow Flare on Cid who had only 6000 hp at that time, therefore killing him. That annoyed me deeply since the Exp and AP (50000 Exp and a couple hundred AP) were greatly needed. I reloaded the game a couple of times, but it seems that it does always cast this spell just before dying (not unlike the Gargoils in the Northern Crater that cast automatically L4 Death just before dying), and very strangely, always on the same character.
The Emerald Weapon
I had Cloud, Cid and Tifa to defeat the Emerald Weapon. All were level 99, had 9999 hp and 999 mp (Hey, you're tough or you're not!). All had also their extreme limit skills, but I did not use more than once omnislash and maybe once the Knights. With the 4x-cut materia, you can inflict a lot of damage! Just limiting the damage the Emerald Weapon does to you makes all the difference. I guess that the Emerald Weapon has about 500000 hp, so about four rounds of 4x-cut attacks and it's dead.
The Ruby Weapon
Another thing is that it seems to me that the Ruby Weapon is invincible as long as it does not bury its claws in the sand, which in turn it only does when there's only one active character in your party (the two other being dead or whirlsanded). After that, well, you have to use:
Since it is relatively impervious to ordinary weapons attacks (4x-cut and the Ultima Weapon inflict about 2000 hp of damage, which sucks) you will have to use the Master Summon materia and call the Knights of Round as many times as you can. Since it has about 800,000 hp, you will summon them about 8 to 12 times depending on how much damage they will inflict. The Hp drain materia will keep you alive, but you will need elixirs and ethers to feed you magic points, because the summonings are expensive (320 points, is it, for Knights of Round?).I had to summon them 9 times to kill the Ruby Weapon. When it dies, you get 50000 exp, 50000 ap, and some gold. You will also get the Desert Rose item.
You will get the Guide Book by morphing the ghost ship monster (that you encounter in the undersea passage to the mako reactor). In exchange of the Guide Book, the old man gives you the underwater materia, which will allow you to survive more than 20 minutes under the sea to fight the Emerald Weapon (which you would not need anyway if you are strong enough, since you can kill the darn thing in less than 15 minutes).
You will obtain the Earth Harp by killing the Emerald Weapon. Old man will give you three master materia for it: Master Summon, Master Command and Master Magic. In my opinion, the Master Command sucks, because it does not include the 4x-cut command, which is, I think, quite essential. Anyway the Master Magic and Master Summon are really nice. You can get other Master Summon, Master Command or Master Magic from the three huge materia at Cosmo Canion.
Upon defeating the Ruby Weapon, you will get the Desert Rose item. In exchange for it, the old man will give you a gold chocobo. Useless? Even if you already have one (you must to get the Knights of Round) this one might surprise you.